Transcendence of Euler's Constant γ

Determine whether the Euler–Mascheroni constant γ is rational, irrational, or transcendental. Not one of the three possibilities is excluded.

Euler introduced the constant in 1735. Known: if γ = a/b then b is astronomically constrained (no partial fraction structure helps); Sondow's series and the exponential integral representations connect it to the Gamma function's behaviour, and even γ(\(e^{-γ}\))-style statements about the Gamma function remain open. Hilbert's seventh fell to Gelfond–Schneider within decades; γ has outlived every technique aimed at it.